This premise might sound like sacrilege. Why Moore and not Connery? Well, let me explain. In simple terms, while Connery created the icon, Moore gave the character longevity.
First, a bit of context. Moore assumed the role of Bond at a point when the series was considered moribund and outdated. George Lazenby, Connery's original successor, had been rejected. Connery had come back on a one-off basis to resuscitate the series with Diamonds Are Forever, which provided only a brief reprieve.
Why did Moore succeed where Lazenby had not? Partially this has to do with the relative talents of the actors, but more crucially this has to do with the material they were given. While the movie is one of the best in the series, the filmmakers behind On Her Majesty's Secret Service failed their inexperienced leading man. They tried to wedge poor George into Connery's shoes and in that unequal contest the poor guy could not help but come off second best. As the first man through the door, Lazenby's 'sacrifice' would help the Bond producers figure out a way to re-cast the character. Roger Moore wound up as the first recipient of this new approach.
With Moore, the production team made the decision to tailor the role to the actor rather than the other way around (as with Lazenby). Rather than merely imitate Connery's sophisticated thug, Moore's Bond was a killer comedian, with a penchant for Cuban cigars, puns and a Lotus Esprit as his vehicle of choice. Where Connery was more comfortable beating the hell out of anyone who got in his way, Moore relied more heavily on gadgets and wit to get out of jams.
While his tenure has become a parlour joke, and a certain section of the fanbase think his self-parody destroyed the character, Moore's importance should not be underestimated. Not only did he give the series a second wind, he proved that the character of Bond could live beyond Connery's portrayal. Most significantly, while the argument could be made that he 'destroyed' the idea of who and what Bond was, he also opened up the character to re-invention -- a key factor in the series's continued success.
Moore's interpretation expanded the character's horizons to virtually limitless variations -- now Bond could be camp and silly, or cold and gritty; he could beat a guy up in an elevator, go into outer space, drive a car underwater or chase a free runner across a construction site. Simply put, without Moore, there is no Dalton, no Brosnan, and no Craig. All of Moore's successors have played very different versions of Bond, and that would not be possible if Moore's iteration had not worked. Love him or hate him, Moore's re-invention of Bond ensured the series would continue, and paved the way for all of the character's future permutations.